When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?

Abstract

For a given topological space X we consider two topologies on the hyperspace F( X) of all closed subsets of X. The Fell topology T F on F( X) is generated by the family { O VK : V is open in X and K ⊆ X is compact} as a subbase, where O VK = { F ϵ F( X): F ∩ V ≠ Ø and F ∩ K = Ø}. The topology T F is always compact, regardless of the space X. The Kuratowski topology T K is the smallest topology on F( X) which contains both the lower Vietoris topology T lV , generated by the family {{ F ϵ F( X): F \\ Φ ≠ Ø}: Φ ϵ F( X)} as a subbase, and the upper Kuratowski topology T uK , which is the strongest topology on F( X) such that upper KuratowskiPainlevé convergence of an arbitrary net of closed subsets of X to some closed set A implies that the same net, considered as a net of points of the topological space (F(X), T uK) , converges in this space to the point A. [Recall that a net 〈 A λ 〉 λ ϵ Λ ⊆ F( X) upper Kuratowski-Painlevé converges to A if ∩{ ∪{A μ: μ ⩾ λ} : λ ϵ Λ} ⊆ A .] The inclusion T F ⊆ T K holds for an arbitrary space X, while the equation T F = T K is equivalent to consonance of X, the notion recently introduced by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant spaces X and Y such that their disjoint union X ⊕ Y is not consonant. In particular, locally consonant spaces need not be consonant.